C₅₃×#13710 #27221222

order	13568
nilpotency class	5
isoclinism	isoclinic class of #13709
automorphism	?
homology	Abelianization (H₁) = C₂³×C₅₃ · Schur multiplier (H₂) = C₂

Aliases

direct product

C₅₃×#13710

Subgroups

	subgroup	G/H
center, socle	C₂×C₅₃	#2226
commutator, Frattini, cosocle	C₂×C₁₆	C₂³×C₅₃
prime radical	C₅₃×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₅₃ ◁ C₂×C₄×C₅₃ ◁ C₄×C₈×C₅₃ ◁ C₄×C₁₆×C₅₃ ◁ G

160 conjugacy classes of subgroups; only the 70 normal subgroups are shown.

normal subgroups only

This group has 160 conjugacy classes of subgroups — too many to draw. Only its 70 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 85
transitive degree	μ_t(G) = 1696

A minimal faithful permutation representation — 3 generators on 85 points:

(2,6)(3,13,15,30)(4,7)(5,17)(9,23,25,32)(10,16)(11,12)(14,31,22,26)(18,21,24,29)(19,28)(20,27)(33,34,35,36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85);(1,2,7,11,8,12,4,6)(3,9,22,29,15,25,14,21)(5,10,19,27,20,28,16,17)(13,24,31,32,30,18,26,23);(1,3)(2,9)(4,14)(5,18)(6,21)(7,22)(8,15)(10,26)(11,29)(12,25)(13,27)(16,32)(17,30)(19,23)(20,24)(28,31)

Conjugacy classes & character table

1961 conjugacy classes. The character table is too large to display.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(13568, 6698);